Final answer:
To sketch the polynomial with four zeros at x = {-3, -1, 2, 4} and a positive leading coefficient of an even degree, we can use the product of linear factors (x + 3)(x + 1)(x - 2)(x - 4) and sketch the graph starting below the x-axis, crossing at each zero, and extending upwards indefinitely.
Step-by-step explanation:
To sketch a polynomial with the specified characteristics, you must consider the following:
- The polynomial is of even degree.
- The polynomial has a positive leading coefficient.
- The polynomial has four real zeros at x = {-3, -1, 2, 4}.
An even-degree polynomial with a positive leading coefficient means the ends of the polynomial will be pointing upwards. We can write a simple polynomial that has these zeros by creating a product of linear factors based on the zeros:
f(x) = a(x + 3)(x + 1)(x - 2)(x - 4)
Here, 'a' is a positive constant because the leading coefficient is positive. This polynomial displays four zeros: at x = -3, -1, 2, and 4. The sketch of this polynomial would start below the x-axis, since the value for x < -3 will lead to a negative product (an even number of negative factors gives a positive result, but the factor (x + 3) will be negative here), cross the x-axis at x = -3, then go above the x-axis, cross it again at x = -1, go below, cross at x = 2, rise above again, and finally cross at x = 4, continuing upwards indefinitely as x becomes large.